As the search for new compounds of a topological insulator (TI) becomes more extensive, it is increasingly important to develop an experimental technique that can identify TIs. In this work, we theoretically propose a simple optical method for distinguishing between topological and conventional insulator thin films. An electromagnetic interference wave consisting of waves transmitted through and reflected by the TI thin film is sensitive to the circular polarization direction of the incident electromagnetic wave. Based on this fact, we can identify a TI by observing the interference wave. This method is straightforward, and thus should propel TI research.
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Over the recent years, the interest in topological insulators (TIs) has been extensively growing [1–3]. One of the reasons for this increase in interest is its potential to exhibit rare properties in electron transport, spin transport, and electromagnetic (EM) response, as well as the fact that it is an ideal testbed where condensed matter and high energy physics are bridged. This novel material exhibits bulk properties similar to those of an conventional insulator, whereas at its surface, it exhibits metallic properties. The latter is ascribed to a peculiar electronic energy spectrum, termed a surface Dirac cone. Material properties originating from this conical dispersion relation are considered to be robust against an external perturbation, since they are protected by symmetries inherent in the system.
A lack of varied experimental methods for identifying TIs could potentially hamper further progress in TI research. A widely used experimental method that can identify TIs is angle resolve photo-emission spectroscopy (ARPES), in which an observation of the Dirac cone located at the Γ point in a surface Brillouin zone signifies the insulator. This powerful method revealed that several insulators, such as Bi2Se3, Bi2Te3, and Sb2Te3, are TIs. These materials, often called second generation materials , maintain time reversal symmetry, and their spin Hall effects attract considerable attention. Although ARPES is full-fledged and well-authorized, it is extensive, and hence, expertise is required for measurement and analysis using ARPES. In this context, a simple scheme that enables us to characterize a topological aspect of a given insulator would be helpful in propelling TI research.
We here theoretically propose a convenient scheme in optics to test TIs: one can distinguish a TI from conventional insulators simply by observing an EM interference pattern, where polarization degrees of freedom of EM waves are fully utilized. The proposed method effectively works in a thin film geometry. The most advantageous aspect of this method over ARPES is that it is a straightforward tabletop method, although it is not intended to specify material parameter values.
Typical results of the TI test are displayed in Figs. 1(a)–1(f). In Figs. 1(a) and 1(b), the signals represent intensity differences ΔI of two EM interference waves as a function of frequency ω. The absence of a signal in Fig. 1(a) indicates that the insulator thin film measured is a conventional one, whereas the finite signal in Fig. 1(b), regardless of the shape, indicates that it is a TI thin film. These TI test results are obtained as described below. Take a conventional/topological thin film sandwiched by other thin films or substrates, as shown in Fig. 2(a). By injecting EM waves with electric polarization from the left and from the right sides, one can observe at the right side an interference of the transmitted wave of and reflected wave of . First, record the intensity of the interference wave |E(+ : +)|2, where E(+ : +) is the sum of the transmitted wave of and the reflected wave of . The results are given in Figs. 1(c) and 1(d). Next, change the polarization of the EM wave from to , and record the intensity of the interference wave |E(+ : −)|2. See Figs. 1(e) and 1(f). Then, subtract the two records ΔI(+ : ±) ≡ |E(+ : −)|2 − |E(+ : +)|2, resulting in Figs. 1(a) and 1(b). As mentioned earlier, the total cancellation of the two plots |E(+ : ±)|2 indicates that it is a conventional insulator thin film, whereas a finite signal indicates a TI, regardless of the signal form. Note that the test results provide us with a qualitative difference, and thus, no further analysis of the signal shape is required.
2. Topological electromagnetic effects
The central idea of the TI test relies on a peculiar property of TIs, namely, the topological EM effect [4–7]. This EM effect is theoretically described by axion electrodynamics. The theory assumes, in addition to the conventional Maxwell equations, an axion term whose Lagrangian form is written as8], it is a quantized material parameter in the context of a TI. The effect of this term on EM properties reveals itself in TIs that have a finite energy gap in the surface Dirac cone due to broken time reversal symmetry. One of the proposed methods for installing the gap in second generation TIs involves a TI surface with a ferromagnetic thin film . The axion term, Eq. (1), maintains a form of the Euler–Lagrange equations derived from the Lagrangian of the conventional Maxwell theory. This is because ℒaxion can be rewritten as a total derivative term, since E · B ∝ ∂μ (εμνρσAν∂ρAσ), where Aμ (μ = 0, 1, 2, 3) is the four component vector potential and εμνρσ is a totally antisymmetric tensor . The effect of the axion term is then incorporated into the framework of classical EM theory by adding extra terms to constitutive relations, thereby resulting in the expression D = εE − atB and H = μ−1B+btE, with permittivity ε and permeability μ. The newly added terms proportional to B(E) in D(H) are hereafter referred to as cross-correlated terms. Although these forms of the constitutive relations are also used in the study of multiferroic materials , the remarkable feature here is that the coefficients of the cross-correlated terms in the TI are identical, or at = bt ∝ θ.
The role of the cross-correlated terms in topological EM responses is interpreted as that of Hall conductivity at the surfaces of the insulator. To verify this, let us consider an interface at z = 0 between a conventional material with ε1 in the semi-infinite region z < 0 and a TI with ε2 in z ≥ 0 [Fig. 2(b)]. The constitutive relations in this case can be expressed as:Eqs. (2) and (3). In the following analysis, we use the Maxwell equation with a current term j = σ⃡E and the conventional forms of the constitutive relations.
3. Optical topological insulator test
To capture the underlying physics in Fig. 1, a use of a free-standing geometry is sufficient, instead of a sandwiched one that is used for obtaining the numerical results in Fig. 1, once effectiveness of the axion term is implicit. Indeed, the essence of the TI test can be read from a manageable form of analytical results in this geometry. Assume that two surfaces of the TI thin film are located at z = 0 and z = d, as shown in Fig. 2(c). Then, the Hall conductivity tensor is of the form12], which directly results from the cross-correlated term bt [Θ(z) − Θ(z − d)] in the corresponding constitutive relation. This leads to topological EM responses that are asymmetric depending on whether the incident EM wave propagates from the right or left side. We now calculate transmittance t and reflectance r of the thin film using the circularly polarized basis E±. Four cases can considered, depending on the pair of the polarization direction of the incident wave and the incident surface. Firstly, consider the case where a plane wave with E+ enters into the thin film from the z < 0 region. The transmittance and reflectance are obtained, respectively, as
A procedure that provides us with an obvious qualitative difference between the two kinds of insulators is devised by exploiting the interference phenomenon of EM waves. To show this, we consider the other three remaining geometries. The relations of each transmittance and reflectance with and are, respectively, given asEq. (10), shows that r depends on the sign of σ̃H. This fact seems to be valuable in devising an optical scheme that enables us to distinguish the insulators. However, the reflectance alone does not meet the current aim, because the quantity that is straightforwardly measured, namely the reflection coefficient R = |r|2, is indeed insensitive to the sign of σ̃H. Therefore, in order to deduce the phase information inherent in r, we use an interference signal of the transmitted and reflected waves. The quantity suitable for our purpose is the subtraction of the two interference waves, defined as Figs. 1(a) and 1(b) for a conventional and topological insulator, respectively. In Fig. 1(b), the fact that r in the TI depends on both the incident wave polarization and the incident surface is observed through the interplay with the transmitted wave, such that ΔI remains finite. However, since both of t and r in conventional insulators are totally symmetric with respect to E±, ΔI(+ : ±) becomes zero. The results obtained in this feasible scheme no longer require further analysis of the shape of ΔI and the two kinds of insulators can be identified without any ambiguity. Although the theoretical foundation of the TI test has been fully presented in the above analytical discussion of a free-standing case, the results shown in Fig. 1 are numerically obtained in a rather realistic sandwiched geometry with using the transfer matrix method, as seen in Fig.2(a), where n1/n0 = 1.5 and n2/n0 = 2.0 are used. Note that the scheme is also effective for an asymmetric geometry.
The heart of the present work is to consider the subtraction between |E(+ : ±)|2. One might think that |E(+ : +)|2 or |E(+ : −)|2 alone is sufficient for identifying a TI, and hence, there is no need to subtract the terms |E(+ : ±)|2, since a finite deferential coefficient d|E(+ : ±)|2/dω|ω=0 would become a subtle indicator of the insulator, as observed in Fig. 1. However, the measurement in D.C. limit using EM waves is far from realistic, and it is practical to rely on A.C. measurement in a frequency range ω ∈ [ωa, ωb].
An appropriate frequency range should be determined from connection with requisite conditions of our model. The effective functioning of our method which relies on Eqs. (7) and (8) assumes that the permitivities ε2 and μ0 are real and insensitive to ω. Therefore, we should avoid the region with a specific spectroscopic structure, e.g., a resonance. To obtain a clear signal, it is also favorable to avoid the region with n2/σ̃H ≫ 1. The frequency range that meets this condition depends on the individual material being studied. Although this careful identification of the frequencies ωa and ωb is required, it is feasible. The frequency selection also depends on a property of magnetic thin films that break time reversal symmetry. In our discussion, the EM properties are assumed to be described by real refractive index n1. This ideal condition would be satisfied by using, for instance, a ferromagnetic thin film that is transparent up to the visible frequency range. Recently, the synthesis of such a material, TiO2 codopoed with Fe and Nb, has been reported . Thus, our intelligible treatment presented in this work gives a firm basis to the TI test. Before closing, we should comment that when our interest is developed into metamaterials, a bianisotropic photonic metamaterial  is found to provide us with ΔI ≠ 0, since constitutive relations in this metamaterial are identical in form with those in topological insulators. In this sense, the metamaterial would fall into a class of topological materials.
In conclusion, we proposed a convenient optical scheme that enables us to distinguish between a topological insulator thin film and a conventional one. Although this method requires a careful selection of the frequency range in which the experiment would be performed, it is quite feasible and straightforward when compared with the current unique method ARPES. The TI test presented in this work will help for experimentalists in this research field to explore new TI compounds.
The author was supported in part by Grant-in-Aid for Scientific Research (C) 22540340 from MEXT, Japan.
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